Arbitrage Theory for Non Convex Financial Market Models

When dealing with non linear trading costs, e.g. fixed costs, the usual tools from convex analysis are inadequate to characterize an absence of arbitrage opportunity as the mathematical model is no more convex. An unified approach is to describe a financial market model by a liquidation value process. This allows to extend the frictionless models of the classical theory as well as the recent proportional transaction costs models to a large class of financial markets with transaction costs including non linear trading costs. The natural question is to which extent the results of the classical arbitrage theory are still valid when the model is not convex, in particular what does the existence of an equivalent separating probability measure mean? Our contribution is a first attempt to characterise the absence of arbitrage opportunity in non convex financial market models